Nonlinear three-dimensional modeling for encapsulated microbubble dynamics subject to ultrasound

Encapsulated microbubbles (EMBs) stabilized by thin coatings have been used as contrast agents for ultrasound sonography as well as having been demonstrated as a promising new technology for targeted drug delivery. The dynamics of EMBs is three-dimensional (3D) because EMBs within micro-vessels inevitably interact with boundaries, but the theoretical and numerical studies are limited to spherical, weakly non-spherical, and/or axisymmetric EMBs. Here, we have developed physical, mathematical, and numerical models for nonlinear 3D EMB dynamics. The liquid flow is evaluated using the boundary integral method. The EMB coating is modeled as a thin viscoelastic shell including stretching, bending, and shear effects and simulated using the finite element method. These models are coupled through the kinematic and dynamic boundary conditions at the interface. The model is in good agreement with the Hoff equation for spherical EMBs and the asymptotic theory for weakly non-spherical deformation of EMBs. Using this model, a numerical study for EMB dynamics near a rigid boundary subject to an ultrasonic wave is performed. The migration, non-spherical oscillation, resonant oscillation, and jetting of EMBs are displayed and analyzed systematically. If the ultrasound wave is strong, a high-speed liquid jet forms at the final stage of the collapse, orientated between the directions of the wave and toward the wall. The EMB jet is weaker and slower and has less momentum, as the non-spherical deformation of the coating and the jetting are suppressed by the viscoelastic property of the coating. If the ultrasound is not strong, the EMB remains spherical for many cycles of oscillation but the EMB undergoes resonant oscillation and becomes significantly non-spherical after several oscillation cycles, when the wave frequency is equal to its natural frequency. The numerical capability has the potential to be developed for the optimization of sonography or drug delivery.